## One loaf of bread and six rolls cost $1.80. At the same prices, two loaves of bread and four rolls cost$2.40. How much does one loaf of bre

Question

One loaf of bread and six rolls cost $1.80. At the same prices, two loaves of bread and four rolls cost$2.40. How much does one loaf of bread cost. I have go the answer which is 90 cents and one roll costs 15 cents but the math Olympiad book solution does not explain it well can someone explain it better?

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6 months 2021-07-15T19:58:02+00:00 1 Answers 23 views 0

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loaf: $0.90; roll:$0.15

Step-by-step explanation:

Let b and r represent the cost of a loaf of bread and a roll, respectively.

The two purchases can be written in equation form as …

b +6r = 1.80

2b +4r = 2.40

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The usual ways to work problems like this are “substitution” or “elimination”. The latter is sometimes called “addition.”

### Substitution

The way these equations are written, we can see that it is relatively easy to write an expression for b from the first equation:

b = 1.80 -6r

Substituting that into the second equation, we get …

2(1.80 -6r) +4r = 2.40

3.60 -8r = 2.40 . . . . . . . collect terms

1.20 = 8r . . . . . . . . . . add 8r-2.40

0.15 = r . . . . . . . . . divide by 8

Then substituting this into the expression for b, we have …

b = 1.80 -6r = 1.80 -6(0.15) = 0.90

One loaf costs $0.90; one roll costs$0.15.

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### Elimination

The idea with “elimination” is to combine the equations in such a way that one of the variables is eliminated in the process. Here, there are several ways that can be done.

We recognize that the b-coefficient in the second equation is double that in the first equation. So, we can subtract the second equation from double the first equation to make the b-terms disappear.

2(b +6r) -(2b +4r) = 2(1.80) -(2.40)

8r = 1.20 . . . . . . . simplify

r = 0.15 . . . . . . . divide by 8

Using the first equation, we can find b:

b + 6(0.15) = 1.80

b = 0.90 . . . . . . . . . subtract 0.90

One loaf costs $0.90; one roll costs$0.15.

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Another way we can eliminate one of the variables is to rewrite the second equation without its excess factor of 2:

b +2r = 1.20

Then subtracting the first equation from 3 times this will eliminate the r variable:

3(b +2r) -(b +6r) = 3(1.20) -(1.80)

2b = 1.80 . . . . . . . . . simplify

b = 0.90

Then, using the above version of the second equation, we have …

0.90 +2r = 1.20

2r = 0.30

r = 0.15

One loaf costs $0.90; one roll costs$0.15.